To find the solutions of \(x^4 - 29x^2 = -100\):
1. Rewrite as \(x^4 - 29x^2 + 100 = 0\).
2. Let \(y = x^2\), turning it into \(y^2 - 29y + 100 = 0\).
3. Factor as \((y - 25)(y - 4) = 0\).
4. Solve for \(y\): \(y = 25\) or \(y = 4\).
5. Since \(y = x^2\), solve for \(x\):
- \(x = \pm 5\) for \(y = 25\).
- \(x = \pm 2\) for \(y = 4\).
6. Real solutions are \(x = -5\), \(x = -2\), \(x = 2\), and \(x = 5\).
No complex solutions exist since all solutions are real.
Therefore, the real solutions are \(\boxed{-5}\), \(\boxed{-2}\), \(\boxed{2}\), and \(\boxed{5}\), with no complex solutions.
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To find the solutions of \(x^4 - 29x^2 = -100\):
1. Rewrite as \(x^4 - 29x^2 + 100 = 0\).
2. Let \(y = x^2\), turning it into \(y^2 - 29y + 100 = 0\).
3. Factor as \((y - 25)(y - 4) = 0\).
4. Solve for \(y\): \(y = 25\) or \(y = 4\).
5. Since \(y = x^2\), solve for \(x\):
- \(x = \pm 5\) for \(y = 25\).
- \(x = \pm 2\) for \(y = 4\).
6. Real solutions are \(x = -5\), \(x = -2\), \(x = 2\), and \(x = 5\).
No complex solutions exist since all solutions are real.
Therefore, the real solutions are \boxed{-5}, \boxed{-2}, \boxed{2}, and \boxed{5}, with no complex solutions.